Finite Group Databasebeta

← #201222

Subgroups — #201222

Order 512.

1093 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60251C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic
5#17404C211yesmaximal, characteristic
6#9452C218yesmaximal, complemented
7#7949C211yesmaximal, characteristic
8#7948C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11#2987—22no—
12#2985—22no—
13#2984—22no—
14#2983C2214yes—
15#2962—22no—
16#2961—22no—
17#2924—22no—
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22#2868C412yes—
23#2862C412yes—
24#2660—22no—
25#2567—22no—
26#2546C2214yes—
27#2502—22no—
28#2434C414yes—
29#2317—22no—
30C4×#117C2211yescharacteristic
31#1558—22no—
32#1558C2214yes—
33#1558C2214yes—
34#1548—22no—
35#1548—22no—
36C2×#333—22no—
37#1472—22no—
38#1470—22no—
39#1467C2214yes—
40#1285C2214yes—
41#1285C2214yes—
42C23×Q8C2×C412yes—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8C2×C412yes—
62C2×C4⋊Q8C2×C418yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2312yes—
66C2×#125D412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8C2312yes—
71C2×#117—21no—
72C2×#117D412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#446—28no—
77#446—24no—
78#446—28no—
79#446—28no—
80C8×Q8—24no—
81#433—28no—
82#432—21no—
83#432—21no—
84#432—21no—
85#432—24no—
86#432—21no—
87#432—21no—
88#432—21no—
89#432—21no—
90#432D412yes—
91#432C2312yes—
92#432C2312yes—
93#432C2312yes—
94#432C2312yes—
95#431—21no—
96#431—24no—
97#431—21no—
98#431—21no—
99#431D412yes—
100#424—24no—
101#423—21no—
102#423D412yes—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D412yes—
105#407—21no—
106#407—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109#405—24no—
110C4×M16—21no—
111C4×M16—21no—
112C2×#97—21no—
113C2×#97—21no—
114#398—28no—
115#395—28no—
116#391—24no—
117#391C2×C418yes—
118#389—28no—
119#385—28no—
120#384—24no—
121#384—24no—
122#382—24no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C22⋊C4—21no—
126C4×C22⋊C4C2×C412yes—
127#376—21no—
128C43C2×C412yes—
129#333—28no—
130#333—28no—
131#324—28no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—84no—
138C22×Q8C22⋊C412yes—
139C22×Q8C22×C414yes—
140C22×Q8C22×C412yes—
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C2×M16—22no—
146C2×M16—41no—
147C22×C8—22no—
148C22×C8—41no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—41no—
166C4⋊Q8—41no—
167C4⋊Q8—41no—
168C4⋊Q8—41no—
169C4⋊Q8—41no—
170C4⋊Q8—41no—
171C4⋊Q8—41no—
172C4⋊Q8—41no—
173C4⋊Q8—88no—
174#125—22no—
175#125—22no—
176#125—22no—
177#125—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194#117—28no—
195#117—22no—
196#117—22no—
197#117—22no—
198#117—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C4⋊C4C22⋊C414yes—
216C2×C4⋊C4C22⋊C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4C22⋊C414yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42C22×C414yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D414yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42#4111yescharacteristic
236C2×C42C2411yescharacteristic
237C4⋊C8—28no—
238C4⋊C8—28no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—41no—
244C4⋊C8—41no—
245C4⋊C8C2×D414yes—
246C4⋊C8C2×D414yes—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255#97—28no—
256#97—22no—
257#97—22no—
258#97—22no—
259#97—22no—
260#97—22no—
261#97—41no—
262C4×C8—28no—
263C4×C8—22no—
264C4×C8—41no—
265#95—22no—
266#95—216no—
267#95—22no—
268C24C2×C22⋊C411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—48no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—84no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8—84no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—84no—
293C22×C4C2×C22⋊C414yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C414yes—
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313M16—44no—
314M16—82no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—82no—
321C4⋊C4—28no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—816no—
343C22⋊C4—28no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C42—28no—
347C42—24no—
348C42—24no—
349C42—44no—
350C42—44no—
351C42C4×D411yescharacteristic
352C42C4×D411yescharacteristic
353C42C4×D411yescharacteristic
354C42C4×D411yescharacteristic
355C42#12011yescharacteristic
356C42#12011yescharacteristic
357C42#12111yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12311yescharacteristic
362C42#12311yescharacteristic
363C42C22×D411yescharacteristic
364C42C22×D411yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—22no—
368C23—22no—
369C23—41no—
370C23#38611yescharacteristic
371C23#38611yescharacteristic
372C23C22×C22⋊C411yescharacteristic
373Q8—44no—
374Q8—82no—
375Q8—82no—
376Q8—82no—
377Q8—84no—
378Q8—84no—
379Q8—84no—
380Q8—84no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4#38611yescharacteristic
410C2×C4#38611yescharacteristic
411C2×C4#38611yescharacteristic
412C2×C4#38611yescharacteristic
413C2×C4#38611yescharacteristic
414C2×C4#38611yescharacteristic
415C2×C4#45614yes—
416C2×C4C2×C4×D411yescharacteristic
417C2×C4C2×C4×D411yescharacteristic
418C2×C4C2×#12011yescharacteristic
419C2×C4C2×#12111yescharacteristic
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C8—44no—
423C8—44no—
424C8—82no—
425C8—82no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—44no—
434C22#188612yes—
435C22C2×#38611yescharacteristic
436C22#288812yes—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—21no—
446C4—48no—
447C4—44no—
448C4—44no—
449C4—84no—
450C4#188612yes—
451C4#190912yes—
452C4C2×#45612yes—
453C2—22no—
454C2—22no—
455C2—41no—
456C2C2×#188611yescharacteristic
457C2#2099011yescharacteristic
458C2C2×#190911yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #9452C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#602512
#352812
#177462
C2×#15582
#174042
#94522
#79492
#79482